These practice problems are not in the text book:

1. Generate the first 11 rows of Pascals Triangle (stop when you’ve reached C(10,10)).
2. Use your answer to number one to find C(8,6) (do not use the formula $latex \binom{8}{6}$).
3. Check that the sum of all of the entries in the last row in your triangle is 2^10. What are you counting when you add the entries in a row of Pascal’s Triangle? (Solution: When you add all of the entries of the last row in your triangle you’re counting all of the subsets of the set {1,2,3,…,10}.)
4. Verify that the number of all subsets of the set {1,2,3,4} is 2^4 either by using Pascal’s Triangle or by writing down all possible subsets.
5. Use multiplication to count all of the subsets of the set {1,2,3,4}. (Hint: for each number between 1 and 4, there are two options: either the number is In the subset or it is Out. For each subset, write down a corresponding sequence of I‘s and O‘s. Now the problem looks just like counting all possible outcomes when we toss a coin 4 times.)
6. Verify that the alternating sum of the last row from your triangle is 0. (The alternating sum means that we alternate between adding and subtracting. For example, the alternating sum of the 4-th row in Pascal’s Triangle is 1-3+3-1.)

The follow problems come from Section 5.6 of your text book:

• 1,2,3,4,9,11,13,15 (For 9-15 odd, shortest possible route means that only South and East steps are allowed.)
• Solutions:
  • 2 Part a: 2^9, Part b: C(9,2)
  • 4 Part a: C(6,0)+C(6,1)+C(6,2), Part b: 2^6-22 (Hint: Think about our solution to number 1 in Quiz 7),

Here are selected solutions for problems not in your text book:

Download (PDF, 186KB)

Here’s a link to download: Selected solutions

Here are some practice problems to help you work through Pascal’s Identity:

From the Chapter 5 (see pages 241-242) project try:

• 1
• 2 We did this at the end of class–this is essentially your extra credit assignment
• 3 This is what we spent most of the class period talking through. Try to write down an explanation, and then check your work with your lectures notes.
• 4
• Challenge problem: 5 part a

The following problems are not in your text book:

• Use Pascal’s Identity to generate C(n,r) for all n and r less than or equal to 8.
• Assuming that C(6,2)=C(6,4), use Pascal’s Identity to show that C(7,2) is equal to C(7,5). Do not use the formula $latex \binom{n}{r}$. (Hint: Convince yourself that C(6,1)=C(6,5)=6.)
• A student group is forming a committee with 3 members. Assuming there are 56 possible ways to choose the members of the committee, how many members does the student group have? (Hint: Use your work from the first of these problems.)

Selected Solutions:

Download (PDF, 571KB)

Here is a link to download: Selected solutions

From Section 5.5 complete: 1-10 odds, 21-25 odds, 29, 31, 35, 37, 49, 61, 63

Practice problems from Section 5.4:

• 1-21 odds only
• 37, 38 (The solution for number 38 is 2^5)
• 49

Practice problems from Section 5.5:

• 27, 34 (solution: 35*34*33*32*31), 39, 51

• Section 5.1 1-13 odds only, 21-26, and 41-47 (do both even and odds for the last ten problems)
• Section 5.2 15-38 odds only (here are examples where you are shading in a region on a Venn Diagram)
• Section 5.3 11-19 (odds only), 26-30, 61-68 (these last four questions are similar to the flag example we did in class)
  • Solutions for Section 5.3:
    • solution to problem 26: 102,
    • solution to problem 28: 52
    • solution to problem 30: 68
    • solution to problem 62: 36
    • solution to problem 64: 56
    • solution to problem 68: 40